Young Women in Harmonic Analysis and PDE

December 2-4, 2016

Judith Campos Cordero (University of Augsburg)

Regularity and uniqueness of minimizers in the quasiconvex case

In the context of integral functionals defined over a Sobolev class of the type $W^{1,p}_g(\Omega,\mathbb{R}^N)$, with $N\geq 1$, the quasiconvexity of the integrand is known to be equivalent to the lower semicontinuity of the functional. In this context, L.C. Evans showed in 1986 that the minimizers are regular outside a subset of their domain of measure zero. On the other hand, E. Spadaro recently provided examples showing that no uniqueness of minimizers can be expected even under strong quasiconvexity assumptions. In this talk we present some results stating that, under the same natural assumptions on the integrand, if the boundary conditions are suitably small, it is possible to obtain full regularity (up to the boundary) for the minimizers and, furthermore, they are unique. This is joint work with Jan Kristensen.